There are 4 balls of different colour and 4 boxes of colours the same as those of the balls. Find the number of ways to put one ball in each box so that only two balls are in boxes with respect to their colour.

Options

A. A) 12
B. B) 6
C. C) 24
D. D) 4

Answer

To solve the problem of placing 4 balls of different colors into 4 boxes of the same colors such that only 2 balls are placed in their corresponding colored boxes, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Balls and Boxes**:
Let's denote the balls as A, B, C, and D, and the boxes as W, X, Y, and Z, where each box corresponds to the color of the respective ball.

2. **Select 2 Balls to Place in Their Corresponding Boxes**:
We need to choose 2 out of the 4 balls to place in their correctly colored boxes. The number of ways to choose 2 balls from 4 is given by the combination formula \( \binom{n}{r} \), which is calculated as:
\[ \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]
So, there are 6 ways to choose which 2 balls will go into their corresponding boxes.

3. **Place the Remaining 2 Balls**:
The remaining 2 balls must be placed in the boxes such that they do not go into their corresponding boxes. This is a derangement problem, where we need to find the number of ways to arrange 2 items such that none of them are in their original position.

For 2 items (let's say balls C and D), there is only 1 way to place them in the boxes (i.e., C goes into the box of D and D goes into the box of C). Thus, the number of derangements (denoted as !n) for 2 items is:
\[ !2 = 1 \]

4. **Calculate the Total Number of Arrangements**:
Now, to find the total number of ways to arrange the balls in the boxes under the given conditions, we multiply the number of ways to choose the 2 balls by the number of derangements for the remaining 2 balls:
\[ \text{Total Ways} = \binom{4}{2} \times !2 = 6 \times 1 = 6 \]

Thus, the total number of ways to place the balls in the boxes such that only 2 balls are in their corresponding boxes is **6**.

### Final Answer:
The total number of ways is **6**.